GMY-33

Brahmagupta

Cyclic quadrilateral.

Reading speed

GoverningBrahmagupta

Governing equation

K=\\sqrt{(s-a)(s-b)(s-c)(s-d)}

$a \in \left[0.2, 12\right]$

where

a: a (m)
b: b (m)
c: c (m)
d: d (m)
K: Brahmagupta (m²)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

K=\\sqrt{(s-a)(s-b)(s-c)(s-d)}

$a \in \left[0.2, 12\right]$

$a = 0.2$ $K = 8.32$

Along the curve

②

Domain of the variable — blue

a_{\min} = 0.2,\quad a_{\max} = 12

$a \in \left[0.2, 12\right]$

$a = 0.2$ $a = 0.2$

Along the curve

③

Response curve — green

K=\\sqrt{(s-a)(s-b)(s-c)(s-d)}

$a \in \left[0.2, 12\right]$

$a = 0.2$ $K = 8.32$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}K}{\mathrm{d}a}

$a \in \left[0.2, 12\right]$

$a = 0.274$ $K^{\prime} = 2.579$

Along the curve

Lecture brief

Historical brief

Pythagoras, Heron, spherical trigonometry and the sphere’s volume are the measuring arts that predate calculus. The sheets compute length, area and angle on plane and sphere. This sheet (GMY-33 — Brahmagupta) is the form associated with Brahmagupta. Working symbols: $a$ , $b$ , $c$ , $d$ $\rightarrow$ $K$ . Cyclic quadrilateral. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

K

from

a

b

c

d

in Geometry via

K=\\sqrt{(s-a)(s-b)(s-c)(s-d)}

Cyclic quadrilateral. Use it when a real geometry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 3.000\,\mathrm{m}

b = 4.000\,\mathrm{m}

c = 5.000\,\mathrm{m}

d = 4.000\,\mathrm{m}

, the governing relation

K=\\sqrt{(s-a)(s-b)(s-c)(s-d)}

yields

K = 15.492\,\mathrm{m^{2}}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a3 mb b4 mc c5 md d4 m

Outputs

Brahmagupta K15.492 m²

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Cyclic quadrilateral. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube